Multiplicity and solution construction in linear elliptic problems with nonlocal nonlinearities
Abstract
This work investigates linear elliptic equations with multiple nonlocal nonlinearities in a bounded domain, which takes the form −∇·(od∇ u) +λfu= η∑j=1khjNj[u]+h0. Here λ and η are positive parameters, and Nj[u] represents a nonlocal term dependent on the unknown solution~u. All coefficients are defined in the context of a wide range of applications. As such equations generally lack a variational structure, a new approach is developed that combines fixed-point arguments with asymptotic techniques. This method establishes the existence and multiplicity of solutions under specific conditions. Of particular interest is the role of the nonlocal nonlinearities, which lead to diverse structures of solutions. To the best of our knowledge, this property is a new finding not typically observed in related nonlocal elliptic problems.
Related
- Existence Results for the Critical Biharmonic Elliptic Equation Involving Singular Nonlinearities
- Anisotropic quasilinear elliptic systems involving nonlinearities with negative critical terms
- Normalized Solutions for a Class of the Nonlinear Elliptic Systems with Mixed-Power Nonlinearities on Bounded Domains
- Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical and subcritical nonlinearities
- Interior Regularity for Stable Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities
BibTeX
@misc{lee2026multiplicity,
title = {{Multiplicity and solution construction in linear elliptic problems with nonlocal nonlinearities}},
author = {Chiun-Chang Lee},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.22295},
eprint = {2606.22295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22295},
}
RIS
TY - UNPB
TI - Multiplicity and solution construction in linear elliptic problems with nonlocal nonlinearities
AU - Chiun-Chang Lee
PY - 2026
DA - 2026/06/21
PB - arXiv
DO - 10.48550/arxiv.2606.22295
UR - https://arxiv.org/abs/2606.22295
AB - This work investigates linear elliptic equations with multiple nonlocal nonlinearities in a bounded domain, which takes the form −∇·(od∇ u) +λfu= η∑_j=1^kh_jN_j[u]+h_0. Here λ and η are positive parameters, and N_j[u] represents a nonlocal term dependent on the unknown solution~u. All coefficients are defined in the context of a wide range of applications. As such equations generally lack a variational structure, a new approach is developed that combines fixed-point arguments with asymptotic techniques. This method establishes the existence and multiplicity of solutions under specific conditions. Of particular interest is the role of the nonlocal nonlinearities, which lead to diverse structures of solutions. To the best of our knowledge, this property is a new finding not typically observed in related nonlocal elliptic problems.
ER -